The bottom line is that your dn is of an age where she can choose whether or not to live in the family home and, if she chooses not to, there's bugger all your SIL and NM can do about it.
From what you've said, SIL was not averse to hearing what DN had to say but NM is obviously a paranoid little Hitler fucker and lost no time getting his hand up SIL's bum and working her mouth, hence all the vitriole she's unjustly thrown at you.
I suggest you let the dust settle before telling SIL what you've said here regarding having been encouraging of the move, which you're sure will be beneficial for all concerned (that'll get her wondering why you're not expressing any regret at her/them going), but, of late, you've also been sympathetic to dn's wish to stay at college with her established pals until she's finished her A levels and that you'd be happy to accomodate her during term time so that her studies are not disrupted by the move.
However if, in the meantime, it transpires that dn is getting the brunt of his angst I suggest she moves in with you sooner rather than later, which may cause SIL to wake up to the fact that NM is a controlling twunt who must have his own way in all things and who doesn't give a shit about her dc.
In a sense this is a battle your dn has to fight with her dm but, as she's only 16 and her dm is being controlled by NM, she's need steadfast allies on her side in the shape of your dh and yourself and, perhaps to a lesser extent, MIL.
It's a shame it's kicked off now, but these things tend to happen for a reason and the reason may be that the gods hidden forces are working to ensure that NM doesn't get a chance to run through SIL's savings and leave her high and dry hundreds of miles from her home.
Fwiw, you're best advised to remain your usual cordial self in your dealings with SIL, but this constraint doesn't apply to your DH who should feel free to give NM her both barrels. How dare they have so little regard for DN and her studies.
If dn does move in with you, as she's under 18 and in full-time education you should apply to your local authority's Children's Services Department to regularise the situation and for financial help towards her upkeep because, if it wasn't for you/dh, they may have to find a foster placement for her with all of the cost that entails. If that is the case, come back for further advice before you approach them.